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Statistics & two-sided sample inference

Mean Confidence Interval Calculator

Compute a bounded Student-t interval from an independent sample's mean, sample SD and size.

180 / MEAN CONFIDENCE INTERVAL

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YOUR CALCULATION

Form a two-sided Student-t interval for an unknown population mean.

Degrees of freedom

9

Two-sided Student-t critical value

2.262

Estimated standard error

1.581

Margin of error

3.577

Lower confidence limit

46.423

Upper confidence limit

53.577

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How the mean confidence interval calculator works

Standard error is sample SD divided by sqrt(n). With n − 1 degrees of freedom, the two-sided Student-t critical value at (1 + confidence fraction)/2 multiplies standard error for margin. Subtract and add that margin to the sample mean.

The quantile uses a bounded incomplete-beta numerical method tested against reference values. Supported sample size is 2 through one million and confidence is 90%, 95% or 99%; zero observed SD collapses the computed interval but does not prove no uncertainty.

Independent, representative observations are required. Exact small-sample coverage assumes an approximately normal population. Confidence refers to long-run repeated-sampling coverage, not a probability assigned to a fixed true mean after this interval is observed.

Worked example

For n = 10, mean 50 and sample SD 5, standard error is about 1.581139. At 95% confidence and nine degrees of freedom, t is about 2.262157 and limits are approximately 46.4232 and 53.5768. Doubling SD to 10 doubles standard error to about 3.162278.

Sources and assumptions

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